Seeking a Advanced Linear Algebra Book (Required Topics)

[bacte2013]

Dear Physics Forum personnel,

I am currently reading the books called "Linear Algebra Done Right" by S. Axler and "Linear Algebra Done Wrong" by S. Treil. On the next semester, I will be taking the "Second Course in Linear Algebra" which will treat the following topics: determinants, dual/quotient/tensor spaces, invariant subspaces, diagonalization, spectral theorem, inner-product spaces, quadratic/canonical forms, groups-rotations/symmetry, basic representation theory, and finite groups. Since the course does not have a required text, I am thinking about choosing on by myself. Could you provide me some names of books that cover those topics in-depth?

 

[Fredrik]

The books you're studying cover most of that (and do it really well). What's missing is representation theory and finite groups. The book "Lie groups, Lie Algebras and representations: An elementary introduction" by Brian C. Hall covers representations really well, but doesn't cover finite groups. I don't know what to recommend for finite groups. I think you can probably use any book on abstract algebra.

Finite groups aren't easier to deal with than infinite groups. You will have to study integers (division algorithm, least common divisor, congruence classes, etc.) and use what you've learned there to prove theorems about finite groups. This should be covered by any book on abstract algebra that covers finite groups.

 

[bacte2013]

The books you're studying cover most of that (and do it really well). What's missing is representation theory and finite groups. The book "Lie groups, Lie Algebras and representations: An elementary introduction" by Brian C. Hall covers representations really well, but doesn't cover finite groups. I don't know what to recommend for finite groups. I think you can probably use any book on abstract algebra.

Finite groups aren't easier to deal with than infinite groups. You will have to study integers (division algorithm, least common divisor, congruence classes, etc.) and use what you've learned there to prove theorems about finite groups. This should be covered by any book on abstract algebra that covers finite groups.

Thank you for the advice! I forgot that both Axler and Treil cover most of topics (I am still at their beginning chapters). I really like Axler's coherent, clean treatment!
After reading both of them, what book should I pick up to learn more about the linear algebra? I am especially interested in the operators, which were not really covered well by my former book Friedberg.

 

[Fredrik]

Linear operators on finite-dimensional vectors spaces are covered really well by both of your books. I don't know if there are any more advanced aspects of that topic that are worth studying, or if those topics are covered by any other book. There are some people here who can give you better advice on this, in particular mathwonk and micromass. The book by Roman is often mentioned in these threads, so you may want to check it out. (If you scroll down to the bottom of the page, you should see a link to a thread about it).

When you know the stuff about linear operators in Axler and Treil, I'd say that the next step is to study inifinite-dimensional vector spaces and linear operators on those. Unfortunately that's an extremely difficult subject. You will need to study topology and some measure and integration theory first. Since the proofs involve topology and the vectors are usually functions, infinite-dimensional vector spaces are considered functional analysis rather than linear algebra, so you won't find the relevant stuff in a book with "algebra" in the title.

I like Axler too, but there are some things that are done better in Treil. In particular determinants. On the other hand, there are some great things in Axler that you won't find in Treil, like the section on polynomials. You mentioned dual spaces and tensors. Treil is the only linear algebra book I know that has a chapter about that.

 

[S.G. Janssens]

For duality as well as a hint of some basic concepts of functional analysis (among lots of other things), you could also have a look at Peter Lax's book.